Semileptonic B -> a_1 l nu decay in QCD

12 Pages·0.15 MB·English

by  T. M. Aliev

Semileptonic B a ℓν decay in QCD
1
→
9
9
9
T. M. Aliev ∗, M. Savcı †
1

n Physics Department, Middle East Technical University
a
J 06531 Ankara, Turkey

4
2

1
v Abstract
5
9
3 The form factors and the branching ratio of the B a ℓν decay are calculated
1
1 →
0 in framework of QCD sum rules. A comparison of our results on form factors and
9
9 branching ratio with the results from constituent quark model is presented.
/
h
p
-
p
e
h
:
v
i
X
r
a
∗e-mail: [email protected]
†e-mail: [email protected]
1 Introduction
Inclusive and exclusive decays of heavy flavors play a complementary role in determina-
tion of fundamental parameters of the Standard Model and of a deeper understanding
the dynamics QCD. Among of all these decays the semileptonic decays occupy a special
place, since their theoretical description is relatively simple and due to the possibility of a
more precise determination of the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements
experimentally.
The new experimental results in studying B meson decays, which have been carried
in recent years, has improved the values of the CKM matrix elements and CP violation
parameter. The experimental prospects at future B factories Belle [1] and BaBar [2], where
manyinclusiveandexclusive channelsareexpectedtobemeasuredmoreprecisely, pushesto
make further analysis on the theoretical side. One of the main goals in these investigations
is a more accurate determination of the CKM matrix element V . For this aim, both
ub
exclusive and inclusive b u transitions will be analyzed. Studying inclusive decays
→
theoretically is easy, but their measurement in experiments are difficult. Using inclusive
decays in determining V , implies the need of using perturbative QCD methods in the
ub
region near the end–point of the lepton spectrum, where many resonances are present and
perturbative results are less reliable. This problem can be avoided by considering exclusive
channels for which measurements in experiments are easy. But unfortunately theoretical
description of the exclusive decays is not as easy as the inclusive decays. This is due to the
fact that, in investigating the exclusive decays there appears the problem of calculating the
hadronic matrix elements, which is directly related to the non–perturbative sector of QCD.
So in an investigation of the exclusive decays, we need non–perturbative methods, such as
QCD sum rules, lattice calculations, etc.
The relevant semileptonic exclusive decays in determining V are B πℓν, B ρℓν
ub
→ →
and B a ℓν. The decays B πℓν and B ρℓν, have been extensively studied
1
→ → →
theoretically in a series of papers, in framework of non–perturbative approaches such as
QCD sum rules [3]–[7], light cone QCD sum rules [8]–[11], lattice calculations [12]–[13],
quark model [14]. Note that the B πℓν and B ρℓν decays have already been observed
→ →
in experiments conducted by the CLEO Collaboration [15]. The semileptonic decay B
→
a ℓν was investigated in framework of the constituent quark model (CQM) in [16]. Yet, this
1
decay has not been observed in experiments that have been conducted so far, but expected
tobedetected infutureB–factories. Forthese reasons, itistherighttimetoinvestigate this
decay in framework of different methods and a comparison of the predictions of different
approaches is necessary. In this work we investigate the semileptonic decay B a ℓν in
1
→
framework of the three point QCD sum rules method.
The present paper is organized as follows. In section 2 we formulate the sum rules for
B a transition form factors. Section 3 is devoted to the numerical analysis of the sum
1
→
rules, and a comparison of our results with the quark model predictions.
1
2 Sum rules for B a transition form factors
1
→
In calculation of the B a transition form factors, we start by considering the following
1
→
correlator
Π (p,p′,q) = i2 d4xd4yeip′x−ipy 0 T d¯(x)γ γ u(x)J (0)¯b(x)γ d(y) , (1)
µν ν 5 µ 5
Z D (cid:12) n o(cid:12)E
(cid:12) (cid:12)
where J = u¯γ (1 γ )b is the weak curren(cid:12)t, with momentum q, d¯γ γ u and(cid:12)¯biγ d are the
µ µ 5 ν 5 5
−
currents with quantum numbers of the final a and initial B mesons with four–momentum
1
p′ and p, respectively.
The correlator is expressed through the invariant amplitudes as follows:
Π = ig Π i(p+p′) p Π iq p Π ε pαp′βΠ + (2)
µν µν 0 µ ν + µ ν − µναβ V
− − − ···
In what follows we calculate the correlator (1) at negative values of p2 and p′2 with the
help of the operator product expansion in QCD, on one side, and saturating (1) by the
lowest meson states in the pseudoscalar and axial vector channels, on the other side. Both
representations areequatedusing Boreltransformationsinp2 andp′2, whichsuppress higher
resonance and continuum contributions, as well as higher dimension operator contributions.
Firstly let us consider the physical part of (1). Saturating (1) by the contributions
coming from B and a mesons, we have
1
1 1
Π (p,p′) = 0 d¯γ γ u a a u¯γ (1 γ )b B B ¯biγ u 0 .(3)
µν p2 m2 p′2 m2 ν 5 1 h 1| µ − 5 | i 5
− B − a1 D (cid:12) (cid:12) E D (cid:12) (cid:12) E
(cid:12) (cid:12) (cid:12) (cid:12)
The weak matrix element B a can b(cid:12)e writte(cid:12)n as (q = p p′) (cid:12) (cid:12)
1
→ −
a (ε,p′) u¯γ (1 γ )b B(p) =
1 µ 5
h | − | i
2A(q2) (p+p′)
ǫ ε∗νpαp′β iε∗(m +m )V (q2)+i(ε∗p) µ V (q2)
m +m µναβ − µ B a1 1 m +m 2
B a1 B a1
2m q
+i(ε∗p) a1 µ V (q2) V (q2) , (4)
q2 3 − 0
h i
where m and ε are the mass and the four–polarization vector of the a meson. Note that
a1 1
form factor V can be related to V and V in the following way
3 1 2
m +m m m
V (q2) = B a1V (q2) B − a1V (q2) , (5)
3 1 2
2m − 2m
a1 a1
and with initial condition
V (0) = V (0) .
3 0
The vacuum–to–meson transition matrix elements are are defined in standard way, namely
m2
B ¯biγ d 0 = f B ,
5 B
m
b
D (cid:12) (cid:12) E
(cid:12) (cid:12) √2m2
0 d¯γ(cid:12) γ u (cid:12)a = a1ε . (6)
ν 5 1 ν
g
D (cid:12) (cid:12) E a1
(cid:12) (cid:12)
(cid:12) (cid:12)
2
Secondly, let us now consider the theoretical part of the sum rules. The theoretical part of
the sum rules is calculated by means of the operator product expansion for the correlator
(1). Up to dimension 6, the operators are determined by the contribution of the bare
loop, and power corrections coming from dimension–3 ψ¯ψ , dimension–4 G2 dimension–
5 m2 ψ¯ψ , and dimension–6 ψ¯ψ 2 operators. Our calcuhlatiions show that thhe ciontributions
com0inhg friom G2 and ψ¯ψ 2h arei negligibly small, and for this reason we don’t take their
h i h i
contribution into consideration in the present analysis.
The double dispersion relation, for the perturbative contribution, can be written as
follows
ρ (s,s′,q2)
Π = dsds′ i +sub. terms , (7)
i (s p2)(s′ p′2)
ZZ − −
where index i describes the necessary invariant amplitudes ((i = 0; +; ; V) (see Eq.
−
(2)), and ρ is the corresponding spectral density. The spectral density is obtained from
i
the usual Feynman integral for the bare loop by replacing
1
2πiδ(p2) .
p2 → −
After standard calculations for the spectral densities, we have
3m s′(s 2m2 s′ +q2)
ρ = b − b − ,
V
4π2 λ3/2
3m s′ 2s′
ρ = b + m4 +sq2 m2(s+q2 s′) ,
0 −8π2 "λ1/2 λ3/2 b − b − #
(cid:16) (cid:17)
3m 1 2s′
ρ = b s′(s′ q2 s+2m2)+ s′ 6m4 +λ+6sq2 6m2(s+q2 s′)
± 8π2 (− λ3/2 − − b λ5/2" b − b −
(cid:16) (cid:17)
3m4(s+s′ q2) λ(s+2m2) 6m2s(s′ +q2 s) 3sq2(s+s′ q2) , (8)
∓ b − ∓ b ∓ b − ∓ − #)
where λ = s2 +s′2 +q4 2sq2 2s′q2 2ss′ is the usual triangle function. For the power
− − −
correction contributions we get
¯
ψψ
(3)
Π = ,
V (p2D mE2)p′2
− b
(m2 q2) ψ¯ψ
Π(3) = b − ,
0 2(p2 mD2)p′2E
− b
¯
ψψ
(3)
Π = ,
± ± 2(p2D mE2)p′2
− b
1 m2 m2 q2
Π(5) = m2 ψ¯ψ + b + b − ,
V − 0 "3(p2 m2)2p′2 2(p2 m2)3p′2 3(p2 m2)2(p′2)2#
D E − b − b − b
3
1 (m2 q2)2 m2 q2
Π(5) = m2 ψ¯ψ + b − + b −
0 − 0 "− 6(p2 m2)p′2 6(p2 m2)2(p′2)2 6(p2 m2)(p′2)2
D E − b − b − b
3m2 4q2 m4 m2q2
+ b − + b − b ,
12(p2 m2)2p′2 4(p2 m2)3p′2#
− b − b
1 m2 m2 q2
Π(5) = m2 ψ¯ψ b b − ,
+ 0 "6(p2 m2)2p′2 − 4(p2 m2)3p′2 − 6(p2 m2)2(p′2)2#
D E − b − b − b
1 m2 m2 q2
Π(5) = m2 ψ¯ψ + b + b − . (9)
− 0 "2(p2 m2)2p′2 4(p2 m2)3p′2 6(p2 m2)2(p′2)2#
D E − b − b − b
In Eqs. (8) and (9) subscripts within the parentheses denote the dimension of the cor-
responding operators. Using Eqs. (3), (4), (6), (7), (8) and (9) and using double Borel
transformation in variables p2 and p′2 and equating two different representations of the cor-
relator (1) we get the following sum rules for the form factors describing B a transition:
1
→
m g (m +m )
A(q2) = b a1 B a1 e(m2B/M12+m2a1/M22)
2√2f m2 m2
B B a1
dsds′ρ (s,s′,q2)e−s/M12−s′/M22 + ψ¯ψ e−m2b/M12
V
× (
ZZ D E
1 m2 m2 q2
m2 ψ¯ψ e−m2b/M12 + b + b − ,
− 0 "− 3M12 4M14 3M12M22#)
D E
m g
V (q2) = b a1 e(m2B/M12+m2a1/M22)
1 √2f m2 m2 (m +m )
B B a1 B a1
m2 q2
dsds′ρ (s,s′,q2)e−s/M12−s′/M22 + ψ¯ψ b − e−m2b/M12
0
× ( 2
ZZ D E
1 (m2 q2)2 m2 q2
+ m2 ψ¯ψ e−m2b/M12 b − + b −
0 "6 − 6M12M22 6M22
D E
3m2 4q2 m4 m2q2
+ b − b − b ,
12M12 − 8M14 #)
m g (m +m )
V2(q2) = b a1 B a1 e(m2B/M12+m2a1/M22)
√2f m2 m2
B B a1
1
dsds′ρ (s,s′,q2)e−s/M12−s′/M22 + ψ¯ψ e−m2b/M12
+
× ( 2
ZZ D E
1 m2 m2 q2
m2 ψ¯ψ e−m2b/M12 + b + b − ,
− 0 "6M12 8M14 6M12M22#)
D E
m g q2
V (q2) V (q2) = b a1 e(m2B/M12+m2a1/M22)
3 0
− 2√2f m2 m3
B B a1
4
1
dsds′ρ (s,s′,q2)e−s/M12−s′/M22 ψ¯ψ e−m2b/M12
−
× ( − 2
ZZ D E
1 m2 m2 q2
+ m2 ψ¯ψ e−m2b/M12 + b + b − . (10)
0 "− 2M12 8M14 6M12M22#)
D E
The region of integration, which appears in calculation of the perturbative contribution, is
determined by the following inequalities:
0 < s′ < s′ ,
0
m2
m2 + b s′ < s < s , (11)
b m2 q2 0
b −
where s and s′ are the continuum thresholds in the B and a meson channels, respectively.
0 0 1
In Eq. (10), the continuum contribution is modeled as the bare loop contribution starting
from the thresholds s and s′ and subtracted from bare loop contribution.
0 0
Finally, using the expressions for the form factors, we present the differential decay
width dΓ/dq2 for the B a ℓν, which can be written in terms of the helicity amplitudes
1
→
λ1/2(m2 ,m2 ,q2)
H (q2) = (m +m )V (q2) B a1 A(q2) ,
± B a1 1 ∓ m +m
B a1
1 λ(m2 ,m2 ,q2)
H (q2) = (m2 m2 q2)(m +m )V (q2) B a1 V (q2) ,
0 2ma1√q2" B − a1 − B a1 1 − mB +ma1 2 #
in the following way
dΓ G2 V 2
± = | ub| q2λ1/2(m2 ,m2 ,q2) H 2 ,
dq2 192π3m3 B a1 | ±|
B
dΓ G2 V 2
0 = | ub| q2λ1/2(m2 ,m2 ,q2) H 2 , (12)
dq2 192π3m3 B a1 | 0|
B
where , 0 refers to the a helicities. Note that the difference of the form factors V V
1 3 0
± −
does not give any contribution to the B a ℓν decay, since it is proportional to the lepton
1
→
mass (in our case electron or muon).
3 Numerical analysis
In calculation of the form factors A(q2), V (q2), V (q2) and V (q2), we use the following
1 2 0
values of the input parameters: m = 4.8 GeV, m = 1.26 GeV, g = 8.9 [17], m =
b a1 a1 B
5.28 GeV, s′ = 3 GeV2, ψ¯ψ = (0.24 GeV)3 [18], m2 = (0.8 0.2) GeV2 [19] (at the
0 − 0 ±
normalization point µ = D1 GeEV). For the value of the leptonic decay constant, we shall
use f = 140 GeV. This value of f corresponds to the case where (α ) corrections are
B B s
O
not taken into account (see [20, 21]). Since the bare loop contribution does not involve
(α ) corrections, they are not taken into consideration in calculation of the leptonic decay
s
O
constant as well, for sake of a consistent procedure.
5
The expressions for the form factors involves two independent parameters M2 and M2.
1 2
According to the QCD sum rules ideology, the problem is to find a region where the results
are practically independent of M2 and M2, while at the same time, the power corrections
1 2
andcontinuum contributions remain under control. Onevidence ofthe existing calculations
(see [22]), we expect the regionof stability for three–point correlator to be at values of Borel
parameters twice as large as in the corresponding two–point functions. From an analysis of
the two–point functions, it follows that the stability region is 4 GeV2 < M2 < 8 GeV2 ((see
1
[20]). Therefore we find it convenient to evaluate in the range 8 GeV2 < M2 < 15 GeV2. In
1
determining the range of the other independent parameter M2, we have used the following
2
relation,
M2 m2 1
2 a1 , (13)
M2 ≃ m2 m2 ≃ 3
1 B − b
from which it follows that M2 M2/3. We have checked our results weak dependence on
2 ≃ 1
this ratio.
InFigs. 1 (a)–(d),we present theM2 dependence of theformfactorsA(q2 = 0), V (q2 =
1 1
0), V (q2 = 0) and V (q2 = 0) at fixed values M2/M2 = 1/3, s′ = 3 GeV2 and at three
2 0 2 1 0
different values of the threshold, s = 33 GeV2, s = 35 GeV2, s = 37 GeV2 . From
0 0 0
these figures we observe that the variations in the results, with changing s , are about
0
5%. It should be noted here that, in all cases of interest, the main contribution comes
∼ ¯
from the dimension 3 operator ψψ . Our final results for the values of the form factors
at q2 = 0 are presented in TableD1, iEn which we present the CQM predictions at q2 = 0 as
well. A comparison of these results yields that the magnitude of the form factor A(q2 = 0)
is approximately five times larger, while the magnitude of the form factors V (q2 = 0) and
0
V (q2 = 0) are five and two times smaller, respectively, in our case, than the ones predicted
1
by the CQM (the errors in form factors predicted by CQM is about 15%, see [16]). The
magnitude of the form factor V (q2 = 0) is practically the same in both approaches. Figs.
2
2 (a)–(d), depict the q2 dependence of the form factors at fixed values M2 = 10 GeV2
1
and M2 = 12 GeV2, under the condition M2/M2 = 1/3, and at s = 35 GeV2, in the
1 2 1 0
range 0 < q2 < 10 GeV2. The reason why we consider this region of q2 is that the non–
perturbative contribution becomes large and hence the operator product expansion breaks
down. In order to extend our results to the full physical region q2 16 GeV2, we
max ≃
have used the following extrapolation formula for the form factors in such a way that they
reproduce the sum rules prediction up to 10 GeV2 region. We have found that the best
fits, which satisfy the above–mentioned condition, can be expressed as
F (0)
F (q2) = i .
i 1 a (q2/m2 )+b (q2/m2 )2
− F B F B
The values of the constant fit parameters a and b , for different form factors, are listed in
F F
Table 1. After integrating Eq. (12) over q2, and using V = 0.032, τ = 1.56 10−12 s, one
ub B
×
can obtain the decay width and branching ratio of the B a ℓν process. These results,
1
→
together with CQM predictions, are summarized in Table 2, which constitute the main
results of this work. From a comparison of these results we see that, the CQM’s prediction
of branching ratio is approximately five times larger than the QCD sum rules prediction.
6
This work a b CQM [16]
F F
A(0) 0.67 0.10 0.72 0.20 0.14
− ± −
V (0) 0.42 0.05 0.44 0.45 0.81
1
− ± −
V (0) 0.53 0.05 0.45 0.13 0.56
2
− ±
V (0) 0.23 0.05 0.86 0.38 1.20
0
− ± −
Table 1:
This work CQM [16]
Γ (B a ℓν) 5.1 105 s−1 (4.6 0.9) 107 s−1
+ 1
→ × ± ×
Γ (B a ℓν) 7.5 107 s−1 (0.98 0.18) 108 s−1
− 1
→ × ± ×
Γ (B a ℓν) 2.7 107 s−1 (4.0 0.7) 108 s−1
0 1
→ × ± ×
(B a ℓν) 1.6 10−4 (8.4 1.6) 10−4
1
B → × ± ×
Table 2:
7
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8
Figure captions
Fig. 1 The dependence of the form factors A(q2 = 0), V (q2 = 0), V (q2 = 0), and
1 2
V (q2 = 0) on Borel parameter M2. In all figures (a)–(d), the dotted line corresponds to
0 1
value of threshold s = 33 GeV2, the solid line corresponds to s = 35 GeV2, and dash–
0 0
dotted line corresponds to s = 37 GeV2, respectively.
0
Fig. 2 The dependence of the form factors A(q2) ,V (q2) ,V (q2) and V (q2) on q2 at
1 2 0
fixed value of threshold s = 35 GeV2. In all figures (a)–(d), the solid line corresponds to
0
thefixedvalueoftheBorelparameterM2 = 10GeV2, andthedash–dottedlinecorresponds
1
to M2 = 12 GeV2, respectively.
1
9